The region represented by
Question: The region represented by $|x-y| \leq 2$ and $|x+y| \leq 2$ is bounded by a :square of side length $2 \sqrt{2}$ unitsrhombus of side length 2 unitssquare of area $16 \mathrm{sq}$, unitsrhombus of area $8 \sqrt{2}$ sq. unitsCorrect Option: 1 Solution: $|x-y| \leq 2$ and $|x+y| \leq 2$ Square whose side is $2 \sqrt{2}$...
Read More →The value of
Question: The value of $\int_{0}^{2 \pi}[\sin 2 x(1+\cos 3 x)] d x$, where $[t]$ denotes the greatest integer function, is :$-2 \pi$$\pi$$-\pi$$2 \pi$Correct Option: , 3 Solution: $I=\int_{0}^{2 \pi}[\sin 2 x(1+\cos 3 x)] d x$ $I=\int_{0}^{\pi}([\sin 2 x+\sin 2 x \cos 3 x]+[-\sin 2 x-\sin 2 x \cos 3 x]) d x$ $=\int_{0}^{\pi}-\mathrm{dx}=-\pi$...
Read More →If the line x - 2y = 12 is tangent to the ellipse
Question: If the line $\mathrm{x}-2 \mathrm{y}=12$ is tangent to the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ at the point $\left(3, \frac{-9}{2}\right)$, then the length of the latus recturm of the ellipse is :9$8 \sqrt{3}$$12 \sqrt{2}$5Correct Option: 1 Solution: Tangent at $\left(3,-\frac{9}{2}\right)$ $\frac{3 x}{a^{2}}-\frac{9 y}{2 b^{2}}=1$ Comparing this with $x-2 y=12$ $\frac{3}{a^{2}}=\frac{9}{4 b^{2}}=\frac{1}{12}$ we get $a=6$ and $b=3 \sqrt{3}$ $\mathrm{L}(\mathrm{LR})=\fr...
Read More →Let Z_1 and Z_2 be two complex numbers satisfying
Question: Let $Z_{1}$ and $Z_{2}$ be two complex numbers satisfying $\left|Z_{1}\right|=9$ and $\left|Z_{2}-3-4 i\right|=4$. Then the minimum value of $\left|Z_{1}-Z_{2}\right|$ is :01$\sqrt{2}$2Correct Option: 1 Solution: $\left|z_{1}\right|=9, \quad\left|z_{2}-(3+4 i)\right|=4$ $C_{1}(0,0)$ radius $r_{1}=9$ $C_{2}(3,4)$, radius $r_{2}=4$ $\mathrm{C}_{1} \mathrm{C}_{2}=\left|\mathrm{r}_{1}-\mathrm{r}_{2}\right|=5$ $\therefore$ Circle touches internally $\therefore\left|z_{1}-z_{2}\right|_{\min ...
Read More →If a cuver passes through the point (1,-2)
Question: If a cuver passes through the point $(1,-2)$ and has slope of the tangent at any point $(\mathrm{x}, \mathrm{y})$ on it as $\frac{x^{2}-2 y}{x}$, then the curve also passes through the point :$(-\sqrt{2}, 1)$$(\sqrt{3}, 0)$$(-1,2)$$(3,0)$Correct Option: , 2 Solution: $\frac{d y}{d x}=\frac{x^{2}-2 y}{x}$ (Given) $\frac{d y}{d x}+2 \frac{y}{x}=x$ $\mathrm{I} F=e^{\int \frac{2}{x} d x}=x^{2}$ $\therefore y \cdot x^{2}=\int x \cdot x^{2} d x+C$ $=\frac{x^{4}}{y}+C$ hence bpasses through $...
Read More →If y = y (x) is the solution of the differential equation
Question: If y = y (x) is the solution of the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}}=(\tan \mathrm{x}-\mathrm{y}) \sec ^{2} \mathrm{x}, \mathrm{x} \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, such that $\mathrm{y}(0)=0$, then $\mathrm{y}\left(-\frac{\pi}{4}\right)$ is equal to:$2+\frac{1}{e}$$\frac{1}{2}-\mathrm{e}$$e-2$$\frac{1}{2}-\mathrm{e}$Correct Option: , 3 Solution: $\frac{\mathrm{dy}}{\mathrm{dx}}=(\tan x-y) \sec ^{2} x$ Now, put $\tan \mathrm{x}=\mathrm{t} \Rightarrow...
Read More →In a game, a man wins Rs.
Question: In a game, a man wins Rs. 100 if he gets 5 of 6 on a throw of a fair die and loses Rs. 50 for getting any other number on the die. If he decides to throw the die either till he gets a five or a six or to a maximum of three throws, then his expected gain/ loss (in rupees) is :$\frac{400}{3}$ gain$\frac{400}{3} \operatorname{loss}$0$\frac{400}{9}$ lossCorrect Option: , 3 Solution: Expected Gain/ Loss $=$ $=\mathrm{w} \times 100+\mathrm{Lw}(-50+100)+\mathrm{L}^{2} \mathrm{w}(-50-50+$ $100...
Read More →If the solve the problem
Question: If $f(x)= \begin{cases}\frac{\sin (p+1)+\sin x}{x} , x0 \\ q x=0 \\ \frac{\sqrt{x+x^{2}}-\sqrt{x}}{x^{3 / 2}} , x0\end{cases}$ is continuous at x = 0, then the ordered pair (p,q) is equal to :$\left(\frac{5}{2}, \frac{1}{2}\right)$$\left(-\frac{3}{2},-\frac{1}{2}\right)$$\left(-\frac{1}{2}, \frac{3}{2}\right)$$\left(-\frac{3}{2}, \frac{1}{2}\right)$Correct Option: , 3 Solution: $\mathrm{RHL}=\lim _{x \rightarrow 0^{+}} \frac{\sqrt{x+x^{2}}-\sqrt{x}}{x^{\frac{3}{2}}}=\lim _{x \rightarro...
Read More →The number of integral values of
Question: The number of integral values of $m$ for which the quadratic expression. $(1+2 m) x^{2}-2(1+3 m) x+4(1+m), x \in R$, is always positive, is : 8763Correct Option: , 2 Solution: Exprsssion is always positve it $2 m+10 \Rightarrow m-\frac{1}{2}$ $\$ $\mathrm{D}0 \Rightarrow \mathrm{m}^{2}-6 \mathrm{~m}-30$ $3-\sqrt{12}\mathrm{m}3+\sqrt{12}$ ......(iii) $\therefore$ Common interval is $3-\sqrt{12}\mathrm{m}3+\sqrt{12}$ $\therefore$ Intgral value of $\mathrm{m}\{0,1,2,3,4,5,6\}$...
Read More →Solve this following
Question: Let $\alpha$ and $\beta$ be the roots of the equation $x^{2}+x+1=0$. Then for $y \neq 0$ in $R$, $\left|\begin{array}{ccc}y+1 \alpha \beta \\ \alpha y+\beta 1 \\ \beta 1 y+\alpha\end{array}\right|$ is equal to$y^{3}$$y^{3}-1$$y\left(y^{2}-1\right)$$y\left(y^{2}-3\right)$Correct Option: 1 Solution: Roots of the equation $x^{2}+x+1=0$ are $\alpha=$ $\omega$ and $\beta=\omega^{2}$ where $\omega, \omega^{2}$ are complex cube roots of unity $\therefore \Delta=\left|\begin{array}{ccc}y+1 \om...
Read More →In a class of 60 students,
Question: In a class of 60 students, 40 opted for $\mathrm{NCC}, 30$ opted for NSS and 20 opted for both NCC and NSS. If one of these students is selected at random, then the probability that the student selected has opted neither for NCC nor for NSS is :$\frac{2}{3}$$\frac{1}{6}$$\frac{1}{3}$$\frac{5}{6}$Correct Option: , 2 Solution: $\mathrm{A} \rightarrow$ opted NCC $B \rightarrow$ opted NSS $\therefore \mathrm{P}($ nither $\mathrm{A}$ nor $\mathrm{B})=\frac{10}{60}=\mathrm{z} \frac{1}{6}$...
Read More →If a directrix of a hyperbola centred at the
Question: If a directrix of a hyperbola centred at the origin and passing through the point $(4,-2 \sqrt{3})$ is $5 x=4 \sqrt{5}$ and its eccentricity is e, then :$4 e^{4}-24 e^{2}+35=0$$4 \mathrm{e}^{4}+8 \mathrm{e}^{2}-35=0$$4 e^{4}-12 e^{2}-27=0$$4 \mathrm{e}^{4}-24 \mathrm{e}^{2}+27=0$Correct Option: 1 Solution: Hyperbola is $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ $\frac{\mathrm{a}}{\mathrm{e}}=\frac{4}{\sqrt{5}}$ and $\frac{16}{\mathrm{a}^{2}}-\frac{12}{\mathrm{~b}^{2}}=1$ $\mathrm{a}^{...
Read More →Let S and S' be the foci of the ellipse and
Question: Let $S$ and $S^{\prime}$ be the foci of the ellipse and $B$ be any one of the extremities of its minor axis. If $\Delta \mathrm{S} ' \mathrm{BS}$ is a right angled triangle with right angle at $B$ and area $\left(\Delta S^{\prime} B S\right)=8$ sq. units, then the length of a latus rectum of the ellipse is :$2 \sqrt{2}$24$4 \sqrt{2}$Correct Option: , 3 Solution: $\mathrm{m}_{\mathrm{SB}} \cdot \mathrm{m}_{\mathrm{SB}}=-1$ $b^{2}=a^{2} e^{2} \quad \ldots .$ (i) $\frac{1}{2} S^{\prime} B...
Read More →Solve this following
Question: Let $\sum_{\mathrm{k}=1}^{10} f(\mathrm{a}+\mathrm{k})=16\left(2^{10}-1\right)$, where the function $f$ satisfies $f(\mathrm{x}+\mathrm{y})=f(\mathrm{x}) f(\mathrm{y})$ for all natural numbers $\mathrm{x}, \mathrm{y}$ and $f(1)=2$. then the natural number ' $a$ ' is 43162Correct Option: , 2 Solution: From the given functional equation : $f(x)=2^{x} \quad \forall x \in N$ $2^{a+1}+2^{a+2}+\ldots .+2^{a+10}=16\left(2^{10}-1\right)$ $2^{a}\left(2+2^{2}+\ldots .+2^{10}\right)=16\left(2^{10...
Read More →If the solve the problem
Question: The sum $\frac{3 \times 1^{3}}{1^{2}}+\frac{5 \times\left(1^{3}+2^{3}\right)}{1^{2}+2^{2}}+\frac{7 \times\left(1^{3}+2^{3}+3^{3}\right)}{1^{2}+2^{2}+3^{2}}+\ldots \ldots .$660620680600Correct Option: 1 Solution: $\mathrm{T}_{\mathrm{n}}=\frac{(3+(\mathrm{n}-1) \times 2)\left(\mathrm{l}^{3}+2^{3}+\ldots+\mathrm{n}^{3}\right)}{\left(1^{2}+2^{2}+\ldots+\mathrm{n}^{2}\right)}$ $=\frac{3}{2} n(n+1)=\frac{n(n+1)(n+2)-(n-1) n(n+1)}{2}$ $\Rightarrow S_{n}=\frac{n(n+1)(n+2)}{2}$ $\Rightarrow S_...
Read More →If the sum of the first 15 tems of the
Question: If the sum of the first 15 tems of the $\operatorname{series}\left(\frac{3}{4}\right)^{3}+\left(1 \frac{1}{2}\right)^{3}+\left(2 \frac{1}{4}\right)^{3}+3^{3}+\left(3 \frac{3}{4}\right)^{3}+\ldots$ is equal to $225 \mathrm{k}$, then $\mathrm{k}$ is equal to :92710854Correct Option: , 2 Solution: $\mathrm{S}=\left(\frac{3}{4}\right)^{3}+\left(\frac{6}{4}\right)^{3}+\left(\frac{9}{4}\right)^{3}+\left(\frac{12}{4}\right)^{3}+\ldots \ldots \ldots \ldots 15$ term $=\frac{27}{64} \sum_{r=1}^{...
Read More →The mean and the variance of five observation
Question: The mean and the variance of five observation are 4 and $5.20$, respectively. If three of the observations are 3,4 and 4 ; then then absolute value of the difference of the other two observations, is:1375Correct Option: , 3 Solution: mean $\overline{\mathrm{x}}=4, \sigma^{2}=5.2, \mathrm{n}=5, . \mathrm{x}_{1}=3 \mathrm{x}_{2}=4=\mathrm{x}_{3}$ $\sum \mathrm{x}_{\mathrm{i}}=20$ $x_{4}+x_{5}=9$ ...........(i) $\frac{\sum x_{i}^{2}}{x}-(\bar{x})^{2}=\sigma^{2} \Rightarrow \sum x_{i}^{2}=...
Read More →Assume that each born child is equally likely to be a boy or a girl.
Question: Assume that each born child is equally likely to be a boy or a girl. If two families have two children each, then the conditional probability that all children are girls given that at least two are girls is :$\frac{1}{11}$$\frac{1}{17}$$\frac{1}{10}$$\frac{1}{12}$Correct Option: 1 Solution: $P(B)=P(G)=1 / 2$ Required Proballity = $\frac{\text { all } 4 \text { girls }}{(\text { all } 4 \text { girls })+(\text { exactly } 3 \text { girls }+\text { lboy })+(\text { exactly } 2 \text { gi...
Read More →The total number of irrational terms in the binomial
Question: The total number of irrational terms in the binomial expansion of $\left(7^{1 / 5}-3^{1 / 10}\right)^{60}$ is :55494854Correct Option: , 4 Solution: General term $\mathrm{T}_{\mathrm{r}+1}={ }^{60} \mathrm{C}_{\mathrm{r}} 7^{\frac{60-\mathrm{r}}{5}} 3^{\frac{\mathrm{r}}{10}}$ $\therefore$ for rational term, $\mathrm{r}=0,10,20,30,40,50,60$ $\Rightarrow$ no of rational terms $=7$ $\therefore$ number of irrational terms $=54$...
Read More →Solve this following
Question: If the line, $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-2}{4}$ meets the plane, $x+2 y+3 z=15$ at a point $P$, then the distance of $P$ from the origin is$\frac{9}{2}$$2 \sqrt{5}$$\frac{\sqrt{5}}{2}$$\frac{7}{2}$Correct Option: 1 Solution: Any point on the given line can be $(1+2 \lambda,-1+3 \lambda, 2+4 \lambda) ; \lambda \in \mathrm{R}$ Put in plane $1+2 \lambda+(-2+6 \lambda)+(6+12 \lambda)=15$ $20 \lambda+5=15$ $20 \lambda=10$ $\lambda=\frac{1}{2}$ $\therefore$ Point $\left(2, \frac{1}{...
Read More →The expression
Question: The expression $\sim(\sim p \rightarrow q)$ is logically equvalent to :$\sim \mathrm{p}^{\wedge} \sim \mathrm{q}$$\mathrm{p}^{\wedge} \mathrm{q}$$\sim p^{\wedge} \mathrm{q}$$p^{\wedge} \sim q$Correct Option: 1 Solution:...
Read More →Prove the following
Question: $\lim _{x \rightarrow 1-} \frac{\sqrt{\pi}-\sqrt{2 \sin ^{-1} x}}{\sqrt{1-x}}$ ie equal to :$\frac{1}{\sqrt{2 \pi}}$$\sqrt{\frac{\pi}{2}}$$\sqrt{\frac{2}{\pi}}$$\sqrt{\pi}$Correct Option: , 3 Solution: $\lim _{x \rightarrow 1^{-}} \frac{\sqrt{\pi}-\sqrt{2 \sin ^{-1} x}}{\sqrt{1-x}} \times \frac{\sqrt{\pi}+\sqrt{2 \sin ^{-1} x}}{\sqrt{\pi}+\sqrt{2 \sin ^{-1} x}}$ $\lim _{x \rightarrow 1^{-}} \frac{2\left(\frac{\pi}{2}-\sin ^{-1} x\right)}{\sqrt{1-x} \cdot\left(\sqrt{\pi}+\sqrt{2 \sin ^{...
Read More →The number of 6 digit numbers that can be formed using the digits 0, 1, 2, 5, 7 and 9 which are divisible by 11 and no digit is repeated, is :
Question: The number of 6 digit numbers that can be formed using the digits 0, 1, 2, 5, 7 and 9 which are divisible by 11 and no digit is repeated, is :36604872Correct Option: , 2 Solution: Sum of given digits 0, 1, 2, 5, 7, 9 is 24. Let the six digit number be abcdef and to be divisible by 11 so |(a + c + e) (b + d + f)| is multiple of 11. Hence only possibility is $a+c+e=12=b+d+f$ Case-I $\{a, c, e\}=\{9,2,1\} \\{b, d, f\}=$ $\{7,5,0\}$ $\{7,5,0\}$ So, Number of numbers $=3 ! \times 3 !=36$ Ca...
Read More →Solve this following
Question: The area (in sq. units) of the region $A=\left\{(x, y): x^{2} \leq y \leq x+2\right\}$ is$\frac{10}{3}$$\frac{9}{2}$$\frac{31}{6}$$\frac{13}{6}$Correct Option: , 2 Solution: $x^{2} \leq y \leq x+2$ $x^{2}=y ; y=x+2$ $x^{2}=x+2$ $x^{2}-x-2=0$ $(x-2)(x-1)=0$ $x=2,-1$ Area $=\int_{-1}^{2}(x+2)-x^{2} d x=\frac{9}{2}$...
Read More →All the pairs (x, y) that satisfy the inequality
Question: All the pairs (x, y) that satisfy the inequality $2 \sqrt{\sin ^{2} x-2 \sin x+5} \cdot \frac{1}{4^{\sin ^{2} y}} \leq 1$ also satisfy the eauation$\sin x=|\sin y|$$\sin x=2 \sin y$$2|\sin x|=3 \sin y$$2 \sin x=\sin y$Correct Option: 1 Solution: $2^{\sqrt{\sin ^{2} x-2 \sin x+5}} \cdot 4^{-\sin ^{2} y} \leq 1$ $\Rightarrow 2^{\sqrt{(\sin x-1)^{2}+4}} \leq 2^{2 \sin ^{2} y}$ $\Rightarrow \sqrt{(\sin x-1)^{2}+4} \leq 2 \sin ^{2} y$ $\Rightarrow \sin x=1$ and $|\sin y|=1$...
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